Alfredo Restrepo

Localized measurement of emergent image frequencies by Gabor wavelets

The authors derive, implement, and demonstrate a computational approach for the measurement of emergent image frequencies. Measuring emergent signal frequencies requires spectral measurements accurate in both frequency and time or space, conflicting requirements that are shown to be balanced by a generalized uncertainty relationship. Such spectral measurements can be obtained from the responses of multiple wavelet-like channel filters that sample the signal spectrum, and that yield a locus of possible solutions for each locally emergent frequency. It is shown analytically that this locus of solutions is maximally localized in both space and frequency if the channel filters used are Gabor wavelets. A constrained solution is obtained by imposing a stabilizing term that develops naturally from the assumptions on the signal. The measurement of frequencies is then cast as an ill-posed extremum problem regularized by the stabilizing term, leading to an iterative constraint propagation algorithm. The technique is demonstrated by application to a variety of 2-D textured images. >

Adaptive trimmed mean filters for image restoration

An adaptive smoothing filter is proposed for reducing noise in digital signals of any dimensionality. The adaptive procedure is based on the selection of an appropriate inner or outer trimmed mean filter according to local measurements of the tail behavior (impulsivity) of the noise process. The set of trimmed means used provides robustness against a wide range of noise possibilities ranging from very shallow tailed to very heavy tailed. A Monte Carlo analysis using a family of generalized exponential distributions supports the choice of the trimmed mean selected for measured values of an impulsivity statistic. The assumption underlying the definition of the filter is that the signal to be filtered is locally smoothly varying, and that the noise process is uncorrelated and derives from an unknown, unimodal symmetric distribution. For image-processing applications, a second statistic is used to mark the location of abrupt intensity changes, or edges; in the vicinity of an edge, the trend-preserving median filter is used. Since the impulsivity and edge statistics used in defining the adaptive filter are both functions of order statistics, the extra computation required for their calculation is minimal. Examples are provided of the filter as applied to images corrupted by a variety of noises. >

Locally monotonic regression

The concept of local monotonicity appears in the study of the set of root signals of the median filter and provides a measure of the smoothness of the signal. The median filter is a suboptimal smoother under this measure of smoothness, since a filter pass does necessarily yield a locally monotonic output; even if a locally monotonic output does result, there is no guarantee that it will possess other desirable properties such as optimal similarity to the original signal. Locally monotonic regression is a technique for the optimal smoothing of finite-length discrete real signals under such a criterion. A theoretical framework in which the existence of locally monotonic regression is proved and algorithms for their computation are given. Regression is considered as an approximation problem in R/sub n/, the criterion of approximation is derived from a semimetric, and the approximating set is the collection of signals sharing the property of being locally monotonic. >

Generalized order statistic filters

The authors explore generalizations of order statistic (OS) filters, a class of finite-width discrete windowing filters defined by linearly weighting the samples in the window according to their natural ordering as real numbers. The concept can be extended by defining generalized ordering rules generating different permutations of the sample prior to weighting. Although the resulting class of generalized OS filter is very broad (including, e.g. the linear filters), local signal ordering properties relating to local signal monotonicity unify them. It is envisaged that both the framework for filter/signal description and the class of filters generated will find use in many signal shaping applications.<<ETX>>

Statistical optimality of locally monotonic regression

We derive the maximum likelihood (ML) estimators for estimating locally monotonic signals embedded in white additive noise, when the noise is assumed to have a density function that is a member of a family of generalized exponential densities with parameter p that includes the Laplacian (p = 1), Gaussian (p = 2) and, as a limiting case, the uniform (p = ∞) densities. The estimators are given by the so-called locally monotonic regression of the noisy signal, a tool of recent introduction in signal processing. The approach that is used in the paper results from a geometric interpretation of the likelihood function of the sample; it takes advantage of the fact that a term in the likelihood function is the p-distance between the vector formed by the data in the given signal (sample) and the vector formed by the elements in the desired signal (estimator). Isotonic regression is a technique used in statistical estimation theory when the data are assumed to obey certain order restrictions. Local monotonicity is a generalization of the concept of isotonicity which is useful for some problems in signal processing.

Spectral properties of moving L-estimates of independent data

Abstract A derivation of the joint probability distribution and mass functions of order statistics coming from overlapping samples is presented. The general formulation allows for samples of any size overlapping (coinciding) in any number of observed values ranging from zero to the number of observations in the smaller sample. These expressions are used to compute the autocovariance function of a moving L -estimate (linear combination of order statistics) of a sequence of independent, identically distributed second-order random variables, under a variety of assumptions on the parent distribution. The associated variance spectral density is also computed for several filters of interest, including median filters, and inner and outer trimmed mean filters.

Optimal noise levels for stochastic resonance

In a stochastic resonance system, additive noise and a nonlinear component system permit the amplification of a weak periodic signal, whenever the strength of the noise is within a certain interval. For one such a system with a nonlinearity consisting of a threshold function, we define a measure of goodness and, for the case of Gaussian noise, derive required intervals of noise variance for stochastic resonance.

Spectral analysis of order statistic filters

We analyze the effect of filter coefficient selection on the power spectral density of an order statistic filtered signal. Assuming that the input signal is a sequence of independent and identically distributed random variates, the autocovariance and the power spectrum of the output are computed. These PSDs are compared with those of the corresponding linear finite impulse response filters with identical coefficients. It is found that, in general, low frequency components predominate regardless of coefficient selection, suggesting an inherent smoothing in the ordering process.

2-D binary locally monotonic regression

We introduce binary locally monotonic regression as a first step in the study of the application of local monotonicity for image estimation. Given an algorithm that generates a similar locally monotonic image from a given image, we can specify both the scale of the image features retained and the image smoothness. In contrast to the median filter and to morphological filters, a locally monotonic regression produces the optimally similar locally monotonic image. Locally monotonic regression is a computationally expensive technique, and the restriction to binary-range signals allows the use of Viterbi-type algorithms. Binary locally monotonic regression is a powerful tool that can be used in the solution of image estimation, image enhancement, and image segmentation problems.

Locally monotonic models for image and video processing

Definitions of locally monotonic images are introduced. The model definitions are complemented with algorithms that compute locally monotonic versions of a given image or video frame input. The property of local monotonicity provides a useful vehicle for image smoothing and denoising. Local monotonicity is also useful for scale space generation, wherein the degree of local monotonicity is the scale parameter. Currently, the property of local monotonicity is well defined for the 1-D case, but is not well defined for images or video. In this paper, models for multidimensional local monotonicity that extend the 1-D definition are rendered. Regression-based and diffusion-based processing methods are prescribed that yield meaningful locally monotonic images. The definitions and associated algorithms are applicable to image enhancement and a variety of multiscale tasks such as image segmentation and video coding.